Collocations Methods For Several Kinds Of Volterra Integral Equations With Delays

Compared with classical integral equations,the delay integral equations are more suit-able to describe those phenomenons with heredity or memory in nature.At present,the delay Volterra integral equations are extensively applied to the fields such as genetics,population models,system control,etc.Also,the related theoretical analysis and numerical methods for solving those equations attract more and more concerns.This doctoral dissertation focuses on collocation methods for several classes of Volterra integral equations with delays,including vanishing and non-vanishing delay Volterra integral equations.For these two kinds of equa-tions,we construct collocation solutions on quasi-geometric meshes and ?-invariant meshes,respectively,and investigate the optimal global and local convergence orders of them.The whole dissertation contains the following five parts:In Chapter 1,we briefly introduce some application background of delay Volterra integral equations.The history and current 'state of the art' of the numerical methods for solving these equations are also listed.In Chapter 2,the collocation methods for vanishing delay Volterra integral equations are discussed.We first construct the collocation methods based on suitable quasi-geometric meshes.Then we focus on the analysis of the optimal global and local convergence orders for the collocation solutions.In addition,we compare these results with the corresponding ones in geometric meshes and uniform meshes.Finally,some numerical experiments are given to verify the theoretical results.In Chapter 3,the non-vanishing delay Volterra functional integral equations are studied.First,the existence,uniqueness,regularity properties,and in particular,the local representa-tions of the exact solutions are discussed.Based on ?-invariant meshes,the global and local convergence properties of the collocation solutions are derived.Finally,we validate the effec-tiveness of the proposed methods by some test examples,and compare the regularity of the exact solutions and convergence orders of the numerical solutions with the ones for general non-vanishing delay Volterra integral equations.In Chapter 4,the vanishing delay Volterra functional integral equations are investigated.By virtue of the perturbation method,we analyze the influence of the initial error on the value of the exact solutions of such equations.With the aid of this result,the relationship of the con-vergence properties of the collocation solutions between vanishing delay equations and their corresponding non-vanishing counterparts is given.Then,we discuss the optimal theoretical convergence orders of the collocation solutions by using the local representation theorem p-resented in Chapter 3.Finally,several representative experiments are shown to confirm the convergence analysis.In Chapter 5,for the purpose of further improving the convergence orders of numerical solutions,the multistep collocation methods are proposed to solve the non-vanishing delay Volterra integral equations.We first construct the multistep collocation solutions for these equations on ?-invariant meshes,and then establish the global and local convergence results of the collocation solutions.Moreover,the comparison of the convergence results between multistep collocation and one step collocation is also given.It is supported by some numerical examples that our methods produce the expected orders.